Contracting the socle in rings of continuous functions

Contracting the Socle in Rings of Continuous Functions

THEMBADUBE(*)

ABSTRACT- Not every ring homomorphism contracts the socle of its codomain to an ideal contained in the socle of its domain. Rarer still does a homomorphism contract the socle to the socle. We find conditions on a frame homomorphism that ensure that the induced ring homomorphism contracts the socle of the co- domain to an ideal contained in the socle of the domain. The surjective among these frame homomorphisms induce ring homomorphisms that contract the socle to the socle. These homomorphisms characterizeP-frames as thoseLfor which every frame homomorphism withLas domain is of this kind.

1. Introduction.

The socle of a ring, it will be recalled, is the ideal generated by the minimal ideals of the ring. In [14] the authors show that, for a completely regular Hausdorff spaceX, the socle of the ringC(X) is the ideal consisting of all functions which are zero everywhere except on a finite number of points. Their method consists of first describing minimal ideals ofC(X).

Because our main aim is to find conditions on a frame homomorphism h:L!Mwhich ensure that the induced ring homomorphismRh:RL!

! RM contracts the socle to an ideal contained in the socle, we take a different approach in characterizing the socle of the ringRLof real-valued continuous functions on a completely regular frameL. What we do is first give a description of annihilators of elements ofRL(Lemma 3.1), and then use the following characterization of the socle

x2SocA,Annx is an intersection of finitely many maximal ideals, to show that the socle ofRLis the ideal consisting of functions whose cozero elements are finite joins of atoms ofL(Proposition 3.5). The spatial result is (*) Indirizzo dell'A.: Department of Mathematical Sciences, University of South Africa, P. O. Box 392, 0003 Unisa, South Africa.

E-mail: dubeta@unisa.ac.za

then obtained as a corollary. We point out how else Proposition 3.5 could have been proved if the sole aim had only been to do that.

Recall [9] that a ring homomorphismf:A!Bis said to be exoteric if whenevera pairof finitely generated ideals ofAhave the same annihilator, then theirimages underfhave the same annihilator. An ideal is then said to be exoteric if it is the kernel of some exoteric homomorphism. We show, using a convenient characterization in [9], that SocRL is exoteric (Pro- position 3.8). In the case thatLhas a finite numberof atoms, we actually produce a ring homomorphism with domainRLwhose kernel is SocRL.

Every frame homomorphismh:L!Minduces a ring homomorphism Rh:RL! RMwhich sends an elementaofRLto the compositeha. In the last section we identify a certain class of frame homomorphisms (we call them W-maps)h:L!Mforwhich (Rh) ¹[SocRM]SocRL(Proposition 4.6). Ifhis a dense onto W-map, then, in fact, (Rh) ¹[SocRM]ˆSocRL.

These homomorphisms are defined in terms of the frame version of the Stone extension of a continuous map. An internal characterization free of the Stone-CÏech compactification is presented (Lemma 4.3). A ring-theoretic characterization is that h is a W-map if and only if Rh contracts every maximal ideal to a maximal ideal (Proposition 4.9). Rather unexpectedly, W- maps characterizeP-frames (i.e. frame counterparts ofP-spaces) as thoseL for which every frame homomorphism with L as domain is of this kind (Proposition 4.4).

An example of a dense onto frame homomorphismhforwhichRhis not an isomorphism but (Rh) ¹[SocRM]ˆSocRL is given (Example 4.5).

Also, an example of a dense onto frame homomorphismh:L!Mwhich is not a W-map but forwhich (Rh) ¹[SocRM]ˆSocRLis given (Example 4.8).

2. Notation and a few reminders.

For a general theory of frames we refer to [13]. Here we collect a few facts that will be relevant for our discussion. Aframeis a complete latticeL in which the distributive law

a^_ Sˆ_

fa^xjx2Sg

holds foralla2LandSL. We denote the top element and the bottom element of L by 1 and 0 respectively. The frame of open subsets of a topological space X is denoted by OX. All spaces are assumed to be Tychonoff, that is, completely regular and Hausdorff.

An elementaofLisrather below(orwell below) an elementb, written ab, if there is an elements such that a^sˆ0 and s_bˆ1. On the otherhand,aiscompletely below b, writtenab, if there are elements (x_r) indexed by the rational numbers inQ\[0;1] such thataˆx₀;x₁ˆb andx_rx_s forr5s.Lis then said to beregularifaˆW

fx2Ljxag foreacha2L, andcompletely regularifaˆW

fx2Ljxagforeach a2L. Although we shall assume all frames considered here to be com- pletely regular, there are instances where we state minor results for regular frames, which then hold for completely regular frames since every completely regular frame is regular.

The pseudocomplement of an element a is the element a ˆ

ˆW

fx2Ljx^aˆ0g. An elementais said to be:

(1) complementedifa_aˆ1, (2) denseifaˆ0,

(3) anatomif, foranyx2L, 0xaimpliesxˆ0 orxˆa, (4) apoint(oraprime element) ifa51 andx^yaimpliesxaor ya.

A frame isatomicif below every nonzero element there is an atom. The meet of finitely many dense elements is dense. If c1;. . .;cm are finitely many complemented elements, then c₁^ ^cm is complemented with (c₁^ ^c_m)ˆc₁_ _c_m. The points of any regular frame are pre- cisely those elements which are maximal below the top (see, for instance, [3, page 12]). Modulo the Axiom of Choice (in the form of Zorn's Lemma), any compact frame has enough pointsin the sense that every element is the meet of points above it; with the convention that the empty meet is the top element. It is for this reason, among others, that we embrace AC throughout.

A frame homomorphism is a map between frames which preserves finite meets and arbitrary joins. By aquotient mapwe mean an onto frame homomorphism. For anya2L,#adenotes the setfx2Ljxag, which is a frame in its own right. We then have theopenquotient map L!#a sending any x2L toa^x. A frame homomorphism is called denseif it maps only the bottom element to the bottom element. Ifhis a dense onto frame homomorphism, thenh(a)ˆh(a)foralla. Associated with a frame homomorphismh:L!Mis itsright adjoint h:M!Lgiven by

h(a)ˆ_

fx2Ljh(x)ag:

Recall thathpreserves meets, and henceh(a)h(b) wheneverab.

An idealJofLiscompletely regularif foreachx2Jthere existsy2J such that xy. For a completely regular frame L, the frame of its completely regular ideals is denoted bybL. Foranya2L,#a2bLif and only if a is complemented. The join map bL!Lis a dense onto frame homomorphism, andbL(together with the join map) is referred to as the Stone-CÏech compactificationofL. We denote the right adjoint of the join mapbL!L byrL (dropping the subscript when it is unnecessary), and recall that for anya2LandI2bL:

(1) r(a)ˆ fx2Ljxag, (2) r(a)ˆr(a),

(3) Iˆr((W I)).

Regarding theframe of reals L(R) and the r ing RL of continuous real-valued functionsonL; we use the notation of [2]. It will be recalled that RL is the collection of frame homomorphisms from L(R) into L.

See also [1] for some properties of the ring RL. The cozero map coz:RL!L is given by

cozWˆ_

fW(p;0)_W(0;q)jp;q2Qg ˆW ( 1;0)_(0;1) : Some of its properties that we shall frequently use include the following:

(1 cozgdˆcozg^cozd, (2) coz (g‡d)cozg_cozd,

(3) coz (g‡d)ˆcozg_cozdifg;d0, (4) cozWˆ0 iffWˆ0.

A frame homomorphism h:L!M induces a ring homomorphism Rh:RL! RMwhich sends an elementaofRLtoha. Furthermore, coz (ha)ˆ(ha) ( 1;0)_(0;1)

ˆh a ( 1;0)_(0;1)

ˆh(coza):

Acozero elementofLis an element of the form cozWforsomeW2 RL.

Thecozero partofL, denoted CozL, is the regular sub-s-frame consisting of all the cozero elements ofL. A frame is completely regular if and only if it is generated by its cozero part, in the sense that every element is the join of cozero elements below it. General properties of cozero elements and cozero parts of frames can be found in [5].

Lastly, by ``ring'' we mean a commutative ring with identity. The an- nihilatorof subsetSis denoted by AnnSand that of an elementaby Anna.

By Ann²we mean the annihilatorof the annihilator. An ideal generated by elementsa₁;. . .;am is written asha₁;. . .;ami.

3. Characterizing the socle ofRL.

We start by recalling some relevant facts about M-ideals ofRL that were introduced in [6]. For anyI2bL, the idealM^IofRLis defined by

M^Iˆ fW2 RLjr(cozW)Ig:

Clearly, for anyI;J2bL,M^IˆM^JimpliesIˆJ, andM^I\M^JˆM^{I^J}. It is proved in [6], Proposition 5.1, that an ideal ofRLis maximal if and only if it is of the formM^Iforsomepoint IofbL.

We require a series of lemmas to be able to prove the main result in this section. The first of these gives a description of annihilators of elements of RL. We prove a more general result than is needed.

LEMMA 3.1. Let S RL and put aˆW

fcozaja2Sg. Then AnnSˆM^r(a). In particular, for any W2 RL,AnnWˆM^r((cozW)). Actu- ally, annihilator ideals ofRL are precisely the idealsM^r(b), for b2L.

PROOF. ForanyW2 RLwe have

W2AnnS,Waˆ0 foreacha2S

,cozW^cozaˆ0 foreacha2S ,cozW^aˆ0

,cozWa ,r(cozW)r(a) ,W2M^r(a);

which proves the first part. Now, to prove the second part, letb2L. Put Bˆ fa2 RLjcozabg. Then bˆW

fcozaja2Bg by complete reg- ularity. So, by the first part,M^r(b)ˆAnnB. p REMARK3.2. (1) Recall that an ideal of a ring is said to beessentialif it has nonzero intersection with every nonzero ideal. For reduced rings this is equivalent to saying the annihilatorof the ideal is the zero ideal. The singular idealof a ringAis the ideal

Z(A)ˆ fa2AjAnnais essentialg:

Since an ideal of a reduced commutative ring is essential if and only if its annihilatoris zero, we see from the lemma that Z(RL)ˆ f0g, since W2Z(RL) iff AnnW is essential, iff AnnM^r((cozW))ˆ f0g, iff M^r((cozW))ˆ f0g, iff cozWˆ0, iff Wˆ0.

(2) In [11] a ringAis said to have thecountable annihilator condition (abbreviated cac) if given any countable setfan jn2Ng A, there exists a2Asuch thatT

nAnnanˆAnna.RLhas the cac, forif (g_n) is a sequence in RL andg is an element of RL such that cozgˆW

cozg_n, then, using Lemma 3.1, one shows thatT

nAnng_nˆAnng.

Next, a quick lemma concerning pseudocomplements. The proof is straightforward and therefore not included.

LEMMA3.3. Suppose aˆc^d with d dense. Then aˆc.

Forthe following lemma, note that ifais an atom in a regular frame, thenais complemented. Indeed, by regularity, there exists 06ˆxa. But then, in light of a being an atom, xˆa, which implies aa, whence a_aˆ1.

LEMMA3.4. The following relations between atoms of L and points of bL hold:

(a)If I is a point ofbL, then(W

I)is the bottom element or an atom of L.

(b)If a is an atom of L, then r(a)is a point ofbL.

PROOF. (a) Note first that every point of any frame is either dense or complemented. IfIis dense, thenW

Iis dense, and hence (W

I) ˆ0. Now supposeIis complemented, and consideranyx2Lwith 05x(W

I). Then x6ˆ1, and hence r(x)6ˆ1bL. Since IIˆr((W

I))r(x)6ˆ1bL, it follows fromIbeing a point thatIˆr(x). Consequently,W

Iˆx, and hence xˆ(W

I). Now let 05c(W

I). By complete regularity, there exists 06ˆxc. Thenx_cˆ1. Butxˆcsince, by what we have shown so far, each of these elements equals (W

I). Soxˆc, and hencec_cˆ1, which impliescˆc ˆ(W

I). Therefore (W

I)is an atom.

(b) Because the right adjoint of a frame homomorphism preserves points, it suffices to show that a is a point of L. Let s2L such that a5s1. Then saˆa. But s6ˆa, so sˆ0, implying that

sˆ1. p

In the following proof we shall need to take cognizance of the fact that if

c1;. . .;cm are finitely many atoms andac1_ _cm, thenais a join of

finitely many atoms. For, aˆ(a^c₁)_ _(a^c_m), and each a^c_i is eitherzero orc_i.

PROPOSITION3.5. SocRLˆ fW2 RLjcoz Wis a join of finitely many atomsg.

PROOF. LetW2SocRL. Take finitely many pointsI₁;. . .;I_kofbLsuch that

AnnWˆM^I1\ \M^Ik:

For brevity, write cozWˆa. Therefore, by Lemma 3.1,M^r(a)ˆM^{I1^^Ik}, which impliesr(a)ˆI₁^ ^I_k:We claim thata(W

I₁)_ _(W I_k). LetIdenote the meet of all the complemented elements offI₁;. . .;I_kgand Jthe meet of the dense ones. ThenIis complemented andJdense. Since r(a)ˆI^J, Lemma 3.3 yields r(a)ˆIˆI_i1_ _I_im forsome in- dicesi1;. . .;im. Taking joins yields

aaˆ_

I_i1_ __

I_imˆ _ I_i1

_ _ _ I_im

;

since the join map, being dense and onto, commutes with pseudocomple- mentation; that is, (W

J)ˆW

J foreachJ2bL. It follows from Lemma 3.4(a) thatais a join of finitely many atoms; proving the inclusion.

Conversely, suppose W is an element of RL such that cozWˆ

ˆa₁_ _a_k, where the a_i are finitely many atoms. Then (cozW) ˆ

ˆa₁^ ^a_k;and hence

r((cozW))ˆr(a₁)^ ^r(a_k) sincerpreserves meets. Thus, by Lemma 3.1

AnnWˆM^r((cozW))ˆM^r(ai)\ \M^r(ak):

Now, by Lemma 3.4(b),r(a_i) is a point ofbLforeachi, soW2SocRL. p COROLLARY3.6. Ifh:L!M is onto, then(Rh)[SocRL]SocRM. PROOF. If SocRL is zero, there is nothing to prove. So suppose otherwise and letW2SocRL. By Proposition 3.5, there are finitely many atomsa1;. . .;aminLsuch that cozWˆa1_ _am. Therefore

coz (hW)ˆh(cozW)ˆh(a1)_ _h(am):

We claim that forany atomaofL,h(a) is an atom ofM. Lety2Mbe such thaty5h(a). We must show thatyˆ0. Sincehis onto, there existsx2L such thath(x)ˆy. By regularity,xˆW

fs2Ljsxg. Take anytxin L. Then t_xˆ1. Therefore aˆ(a^t)_(a^x). Since a is an atom,

a^xˆ0 ora^xˆa. The latteris not possible, lest we haveax, whence h(a)h(x)ˆy, contrary to the fact thaty5h(a). Thus,a^xˆ0, which implies 0ˆh(a)^h(x)ˆh(a)^yˆy, as required. Therefore coz (hW) is a join of finitely many atoms, and hence (Rh)(W)ˆhW2SocRM. p The spatial result mentioned in the introduction is a corollary of Pro- position 3.5. To see this, note first that, for any T1-spaceX, the atoms of OXare precisely the open singletons. Iff:A!Bis a ring isomorphism, thenf[SocA]ˆSocB. Forany topological spaceX, the ring isomorphism C(X)! R(OX) taking f 2C(X) toW2 RLsuch that, forallp;q2Q,

W(p;q)ˆf ¹[fx2Rjp5x5qg];

has the property that

cozWˆ fx2Xjf(x)6ˆ0g;

the cozero-set of f (see [2]). Therefore, for any f 2C(X), f 2SocC(X) if and only if the cozero set off is a join (inOX) of finitely many atoms. Thus, f 2SocC(X) if and only if it is a union of finitely many open singletons;

which is precisely the spatial result in question.

REMARK 3.7. By Lemma 3.1 and Proposition 3.5, Ann (SocRL)ˆ

ˆM^r(a) where adenotes the join of all atoms ofL. Thus, SocRLis es- sential iff aˆ0, iff a is dense, which, in turn, holds iff every nonzero element ofLhas a nonzero meet with some atom, equivalently,Lis atomic.

A ring homomorphismf:A!Bis calledexoteric[9] if forall pairs (I;J) of finitely generated ideals of A, AnnIˆAnnJ implies Annf[I]ˆ

ˆAnnf[J]. An exoteric ideal is an ideal which is a kernel of an exoteric homomorphism. It is shown in [9], Theorem 2.6, that an ideal I of a ring A is exoteric if and only if for any finite set fa₁;. . .;a_mg I, Ann² P

i Ann²a_i

I. We show that SocRL is exoteric. We need two preliminary results from [7], namely Lemma 2.1 and Proposition 2.2 which, respectively, state:

Ifa;b2 RL andcozacozb, thenais a multiple ofb, and

The maph7!cozh is an isomorphism between the Boolean al- gebra of idempotents of RL and the Boolean algebra of com- plemented elements of L.

PROPOSITION3.8. SocRLis an exoteric ideal.

PROOF. Leta₁;. . .;ambe finitely many elements of SocRL. Foreach i2 f1;. . .;mg find atoms a_i1;. . .;a_ini such that coza_iˆa_i1_ _a_ini: Write the sequence

a11;. . .;a1n1;. . .;am1;. . .amnm

as a₁;. . .;a_k. Find idempotents h₁;. . .;h_k in RL such that a_jˆcozh_j for each j2 f1;. . .;kg. Now, since the join of finitely many complemented elements is complemented, and since idempotents are nonnegative in the

f-ringRL, foreachi2 f1;. . .;mg we have

coza_icozh₁_ _cozh_kcozh₁_ _cozh_kˆcoz (h₁‡ ‡h_k):

Consequently,a_iis a multiple ofh₁‡ ‡h_k, and hencea_i2 hh₁;. . .;h_mi SocRL. Therefore

Ann²a_iAnn²hh₁;. . .;h_ki

ˆAnnM^r(a1__ak) by Lemma 3:1

ˆM^r(a1__ak)

ˆM^r(a1__ak);

the last equality since a₁_ _a_k is complemented. But M^r(a1__ak) hh₁;. . .;h_mi, forifW2M^r(a1__ak), thenr(cozW)r(a1_ _ak), so that

cozWcoz (h₁‡ ‡h_k)coz (h₁‡ ‡h_k):

It therefore follows thatP

i (Ann²a_i) hh₁;. . .;h_mi, whence Ann² X

i

(Ann²a_i) hh₁;. . .;h_mi SocRL;

as required. p

In the case that a frame has only finitely many atoms, we can actually identify a ring homomorphism whose kernel is the socle in question. We need a bit of background.

Recall [1] that an onto frame homomorphismh:L!M is called aC- quotient mapprecisely when the ring homomorphismRhis onto. Also,his said to becoz-ontoif foranyc2CozMthere exists d2CozLsuch that h(d)ˆc. Lastly,hisalmost coz-codenseif foranyc2CozLwithh(c)ˆ1, there existsd2CozLsuch thatc_dˆ1 andh(d)ˆ0. It is shown in [1],

Theorem 7.2.7, that an onto homomorphism is aC-quotient map if and only if it is coz-onto and almost coz-codense. Next, recall thathisnearly openif it commutes with pseudocomplements. In particular, if a2L and g:L! #ais the open quotient mapx7!x^a, thengis nearly open. Dense onto homomorphisms are also nearly open.

Now ifcis a complemented element ofL, then the open quotient map g:L! #a is aC-quotient map. That it is coz-onto follows from [1], Cor- ollary 3.2.11, since c is a cozero element. To see that it is almost coz-co- dense, let u2CozL such that g(u)ˆ1_#c. Then u^cˆc, so that cu.

Now c is a cozero element of Lsuch that u_cˆ1 (asc_c ˆ1) and g(c)ˆ0.

LEMMA 3.9. Let h:L!M be a nearly open frame homomorphism.

ThenRh is exoteric.

PROOF. LetIˆ ha₁;. . .;a_miandJˆ hb₁;. . .;b_kibe a pairof finitely generated ideals of RL such that AnnIˆAnnJ. For brevity, write coza_iˆa_i, cozb_iˆb_i, a1_ _am ˆa and b1_ _b_kˆb. Then, WfcozWjW2Ig ˆa, andW

fcozWjW2Jg ˆb. Thus, by Lemma 3.1, M^rL(a)ˆAnnIˆAnnJˆM^rL(b);

so thataˆb. Now, a simple calculation shows that _fcoztjt2(Rh)[I]g ˆh(a);

and _

fcozt jt2(Rh)[J]g ˆh(b):

Thus,

Ann (Rh)[I]ˆM^rM(h(a)) and Ann (Rh)[J]ˆM^rM(h(b)):

By the condition onh,h(a)ˆh(b)sinceaˆb; soRhis exoteric. p PROPOSITION 3.10. Suppose L has a finite number of atoms, and denote by athe join of atoms ofL. Leth:L! #a be the open quotient map. ThenSocRLˆkerRh. Furthermore,RL=SocRL R(#a).

PROOF. Forany a2 RL, a2kerRh iff haˆ0 iff h(coza)ˆ0 iff coza^aˆ0 iff cozaaiff cozais a join of atoms iffa2SocRL. Forthe latterpart, simply note thatRhis onto ashis aC-quotient map as observed earlier, and then invoke the first isomorphism theorem. p

REMARK3.11. We stated in the introduction that a different proof of Proposition 3.5 is available. Here it is: By Brauer's proposition (see [12], Proposition 1 on page 62), a minimal ideal of RL is generated by an idempotent sinceRLis reduced. Now lethbe an idempotent such thathhi is a (nonzero) minimal ideal. We claim that cozh is an atom. Letc be a cozero element ofL(say,cˆcozg) such that 05ccozh. The minimality ofhhiimplieshgi ˆ hhi, so thatcˆcozh. Hence, by complete regularity, if xisanyelement ofLsuch that 05xcozh, thenxˆcozh. So, foreach W2SocRL, cozWis below a join of finitely many atoms, and is therefore a join of finitely many atoms. This establishes the inclusion in the pro- position.

For the reverse inclusion, letabe an atom ofL, and (by the second of the results cited from [7] and stated in the paragraph preceding Proposi- tion 3.8) pick an idempotent jsuch that aˆcozj. Forany a2 RL such that aj6ˆ0, we have 05cozajcozj, whence cozjcozjˆcozaj.

Thus, forsome d2 RL, jˆdajˆdjaj, showing that hjiis a division ring, and hence, by Proposition 2 in [12] on page 63,hjiis a minimal ideal.

So if, forany W2 RL, cozW is a join of finitely many atoms, then cozWcoz (z₁‡ ‡z_m) forsome idempotents z_i each generating a minimal ideal. HenceWis a multiple ofz₁‡ ‡z_m, and is therefore in the socle.

4. Contracting the socle.

We observed in the previous section that a ring homomorphism induced by an onto frame homomorphism extends the socle to an ideal contained in the socle. What we aim to do here is identify certain types of frame homomorphisms for which the induced ring homomorphism contracts the socle to an ideal contained in the socle; that is, homomorphismsh:L!M forwhich (Rh) ¹[SocRM]SocRL. We shall say a frame homomorphism is strongif it satisfies this property. In light of Corollary 3.6, surjective strong homomorphisms have the property that (Rh) ¹[SocRM]ˆSocRL.

We start by giving an example of a frame homomorphism which is not strong. Our example is spatial, so it should be read from the bottom up to keep in line with our frame-theoretic stance.

EXAMPLE 4.1. Endow N[ f0g with the discrete topology, and let XˆYˆN[ f0g. Let h:X!Y be the continuous function defined by h(1)ˆ1 andh(x)ˆ0 forall 16ˆx2X. Next, letgbe the element ofC(Y)

given byg(y)ˆyforally2Y. Note thatg2= SocC(Y). Letf:C(Y)!C(X) be the ring homomorphism f7!fh. We show that g2f ¹[SocC(X)].

Indeed, ifx6ˆ1, then

f(g)(x)ˆ(gh)(x)ˆg(h(x))ˆ0;

so thatf(g)2SocC(X), and henceg2f ¹[SocC(X)]. Thusf ¹[SocC(X)]6 6SocC(Y).

It is well-known that the socle is invariant under an endomorphism.

Since the homomorphismfin Example 4.1 is, in fact, an endomorphism of C(X), the example shows also that the socle is not necessarily ``inverse- invariant'' under endomorphisms.

Now we identify certain types of strong frame homomorphisms. Recall that every homomorphismh:L!M between completely regular frames lifts to the respective Stone-CÏech compactifications as indicated in the following diagram:

The map h^b is given by h^b(I)ˆ fx2Mjxh(y) forsome y2Ig. A straightforward diagram-chasing argument shows thath^b is dense if and only ifhis dense.

DEFINITION4.2. A frame homomorphismh:L!Mis aW-mapif, forall c2CozL,h^br_L(c)ˆr_Mh(c).

These homomorphisms are frame analogues of what Woods calls ``WN- maps'' in [15] defined foronto continuous functions f:X!Yby requiring that, foreach cozero setFofY, cl_bXf ¹[F]ˆ(f^b) ¹[cl_bYF], wheref^bis the Stone extension f^b:bX!bY of f.

We will show that dense W-maps are strong. If Rh were an iso- morphism for every dense W-map h, then of course the result would be trivial. So we start by giving an example of a dense W-map for whichRhis not an isomorphism. Our example will be facilitated by the following characterization of W-maps.

LEMMA4.3. A frame homomorphism h:L!M is a W-map iff for every c2CozL and y2M, yh(c)implies yh(s)for some sc.

PROOF. Lethbe a W-map,ybe an element ofMandca cozero element ofLsuch thatyh(c). Theny2rMh(c)ˆh^brL(c). Thereforeyh(s) for somes2r_L(c).

Conversely, observe first that, for anyc2CozL, in fact, foranyc2L, the inclusion h^br_L(c)r_Mh(c) always holds since, forany s2L, sc implies h(s)h(c). Forthe otherinclusion, let y2rMh(c). Then yh(c), and so, by hypothesis, yh(s) forsome s2rL(c), that is, y2h^br_L(c). This establishes the desired inclusion. p Some more background. TheBooleanizationof a frameLis the Boo- lean frameBLwhose underlying set isBLˆ fxjx2Lgwith meet as in Land join (W

S) foreachSBL. The mapL!BLwhich sends each x2Ltox is a dense onto frame homomorphism. We denote it by[.

Our example will be based on special types of frames. The reader will recall that a P-frame is a frame each of whose cozero elements is com- plemented, and that a frame L is extremally disconnected (or strongly projectable) ifx_x ˆ1 foreach x2L. See [8] for several character- izations ofP-frames. However, we shall need the following characterization (which does not appearin [8]) in terms of W-maps.

PROPOSITION 4.4. The following are equivalent for a completely regular frameL:

(1)L is a P-frame.

(2)Every frame homomorphism h:L!M is a W-map.

(3)[:L!BL is a W-map.

PROOF. (1))(2): Letc2CozLandy2Msuch thatyh(c). Since yh(c) andcc, asLis aP-frame, it follows from Lemma 4.3 thathis a W-map.

(2))(3): Trivial

(3))(1): Let c2CozL. To avoid ambiguity, write forthe com- pletely below relation inBL. Now,[(c)[(c) implies there existsdc in L such that [(c)[(d), by Lemma 4.3. Notice that [(c)[(d) means cd, and dc implies d c. Thereforeccd c;which

impliescis complemented. p

EXAMPLE 4.5. Let X be a P-space which is not extremally discon- nected. Such a space does exist (see [10], 4N). LetLˆOX. ThenLis aP- frame which is not extremally disconnected. We claim that [:L!BLis

not coz-onto. Suppose, by way of contradiction, that[is coz-onto. Letx2L.

Thenx 2CozBL, sinceBLis Boolean. Therefore there existsc2CozL such that[(c)ˆx, that is,cˆx. Sincecˆcandcis complemented, it follows thatx_xˆ1. But this makesLextremally disconnected since x is arbitrary. It follows therefore that [ is not a C-quotient map, and consequentlyR[is not an isomorphism. Thus,[is a dense (onto) W-map for which the induced ring homomorphism is not an isomorphism.

Now we show that ifhis a dense W-map, thenRhcontracts the socle to an ideal contained in the socle.

PROPOSITION4.6. A dense W-map is strong.

PROOF. Let h:L!M be a dense W-map. Let a2(Rh) ¹[SocRM].

Thenha2SocRM. We shall use subscripts to indicate where anM-ideal resides. Take finitely many pointsI₁;. . .;I_kofbMsuch that

Ann (ha)ˆM^I_M¹ \ \M^I_M^k:

Foreachi2 f1;. . .;kg,h^b(I_i) is a point ofbL. We claim that AnnaˆM^h_L^b(I1)\ \M^h_L^b(Ik):

LetW2Anna:Then

(hW)(ha)ˆ(Rh)(W)(Rh)(a)ˆ(Rh)(Wa)ˆ0;

and hence hW2Ann (ha). Therefore r_M(coz (hW))I_i, foreach i2 f1;. . .;kg: But coz (hW)ˆh(cozW); so, foreach i we have rM(h(cozW))I_i. Thus, h^brL(cozW)I_i foreach i since h^brLrMh always. Consequently, r_L(cozW)h^b(I_i), that is, W2M^h_L^b(Ii) foreach i.

Therefore

Anna M^h_L^b(I1)\ \M^h_L^b(Ik):

To show the otherinclusion, let t2M^h_L^b(Ii), foreach i2 f1;. . .;kg. Then r_L(cozt)h^b(I_i), and henceh^br_L(cozt)h^bh^b(I_i)I_i:Sincehis a W-map, this implies

r_M(h(cozt))I_i;

that is,rM(coz (ht))I_i, so thatht2M^I_Mⁱ. As this holds foreachi, we deduce thatht2Ann (ha). Consequently,

0ˆcoz (ht)(ha)

ˆcoz (ht)^coz (ha)ˆh(cozt^coza)ˆh(cozta);

which implies coztaˆ0 by density. Thus,taˆ0, and hence the reverse

inclusion holds. p

In light of Corollary 3.6, we have the following:

COROLLARY 4.7. If h:L!M is a dense onto W-map, then (Rh) ¹[SocRM]ˆSocRL.

Lest one conjecture falsely, we give an example of a dense onto frame homomorphism h:L!M forwhich (Rh) ¹[SocRM]ˆSocRL, buth is not a W-map.

EXAMPLE4.8. LetLˆORand considerthe map[:L!BL. Neither LnorBLhas atoms, so each of the ringsRLandR(BL) has zero socle.

Since[is dense,R[is one-to-one (see [4], Lemma 2). It follows therefore that (R[) ¹[SocR(BL)]ˆ f0g ˆSocRL. However, by Proposition 4.4,[is not a W-map.

A closer look at the proof of Proposition 4.6 shows that what makes it go through, apart from the density of h, is the fact that Rh contracts any maximal ideal to a maximal ideal. As it turns out, this property actually characterizes W-maps as we now show.

PROPOSITION 4.9. A homomorphism h:L!M is a W-map iff (Rh) ¹[M^I_M]ˆM^h_L^b(I)for each pointIofbM.

PROOF. ((): Letc2CozL. Since we always haveh^brL(c)rMh(c), we need to show thatr_Mh(c)h^br_L(c). Ifh^br_L(c)ˆ1_bM, there is nothing to prove. So suppose h^br_L(c)6ˆ1_bM and let I be a point of bM such that h^br_L(c)I. Takeg2 RLsuch thatcˆcozg. Thenr_L(cozg)h^b(I), which implies g2M^h_L^b(I). So, by hypothesis, g2(Rh) ¹[M^I_M], that is, (Rh)(g)ˆ

ˆhg2M^I_M. Thus,

IrM(coz (hg))ˆrM(h(cozg))ˆrMh(c):

Now, since every element ofbMis the meet of points above it, it follows that r_Mh(c)^

fJ2bMjJ is a point aboveh^br_L(c)g ˆh^br_L(c):

()): Leta2(Rh) ¹[M^I_M]. Thenha2M^I_M, and hence r_M(coz (ha))ˆr_M(h(coza))ˆh^br_L(coza)I:

Thus, rL(coza)h^b(I). That is, a2M^h_L^b(I), and therefore (Rh) ¹[M^I_M] M^h_L^b(I). For the reverse inclusion, let W2M^h_L^b(I). Therefore r_L(cozW) h^b(I), which implies

h^br_L(cozW)h^bh^b(I)I;

and consequently, by hypothesis,

r_M(coz (hW))ˆr_Mh(cozW)I:

Therefore hW2M^I_M; that is, (Rh)(W)2M^I_M, implying W2(Rh) ¹[M^I_M].

This proves the reverse inclusion. p

REMARK 4.10. We proved Proposition 4.6 directly. We could, after observing the last proposition, have deduced it as a corollary of the fol- lowing ring-theoretic observation: Iff:A!Bis a one-to-one ring homo- morphism such thatf ¹[M] is a maximal ideal ofAforeach maximal ideal MofB, thenf ¹[SocB]SocA. To see this, leta2f ¹[SocB], and select maximal ideals M₁;. . .;M_k of B such that Annf(a)ˆM₁\ \M_k. One checks easily, using the fact that f is one-to-one, that Annaˆ

ˆf ¹[M1]\ \f ¹[M_k], so that a2SocA.

W conclude with the following comment. If we strengthen the defi- nition of W-map and define anN-mapto be a homomorphismh:L!M such that h^brL(a)ˆrMh(a) forevery a2L (instead of just the cozero elements), then it turns out thatRhis an isomorphism wheneverhis a dense onto N-map. To show this, we first demonstrate thath^b is onto.

Given a2M take b2L such that h(b)ˆa. Then h^b(r_L(b))ˆr_Mh(b)ˆ

ˆr_M(a). Since the elements r_M(x), for x2M, generate bM, it follows thath^b is onto. As remarked earlier, the density ofhimplies that ofh^b. So, in fact,h^b is an isomorphism since it is also one-to-one as any dense frame homomorphism with regular domain and compact codomain is one-to-one. Now we argue from this that h is coz-onto. Let c2CozM.

Since bM!M is coz-onto ([5], Corollary 5), there exists J2CozbM such that W

Jˆc. Since h^b is coz-onto (as it is an isomorphism), there existsI2CozbL such thath^b(I)ˆJ. Then W

Iis a cozero element ofL mapped to c by h. Finally, we show that h is almost coz-codense. So, supposeh(c)ˆ1 for somec2CozL. Then, in view ofhbeing an N-map, h^br_L(c)ˆr_Mh(c)ˆ1_bM, which implies r_L(c)ˆ1_bL since h^b is one-to-one, and hence cˆ1. Thus, h is a C-quotient map, and hence Rh is an iso- morphism.

A similarargument, taking into account the fact that the elements rM(c), forc2CozL, generatebM, shows that ifhis a coz-onto dense W- map, thenRhis an isomorphism.

Acknowledgments. I thank the referee most heartily for helpful re- marks and observations that have strengthened the paper.

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Manoscritto pervenuto in redazione il 22 settembre 2008.